There was no one breakthrough that made it possible. The "big deal", in the mind of the world, was just that an obviously very hard thing was accomplished. And, if you doubted how hard it was, people can point out that no one has done it again in more than fifty years.
However, there are some good examples of challenging problems that had to be solved.
Problem 1: Rocket Size. Before Apollo, everyone thought we would send the top of a multi-stage rocket to the moon, it would land on its tail and launch again to return to earth. When you run the numbers on that, you end up with a pretty big lander requiring a lot of fuel, and a huge launcher to send it on its way; much larger than Saturn. The trick ended up being to only send down a little bug, and even leave part of that behind on the moon. If we had stuck with the giant lander we would never have been ready in time.
Problem 2: Rendezvous. The new method required being good at approaching and docking with another spacecraft. That's a hard enough problem that, even though the physics was well understood, they didn't really see the issues until they actually tried it. (I always get annoyed when characters in science fiction stories fail to foresee problems that the science should have told them beforehand, but sometimes that's how it works.) Wisely they tried it in Gemini in low earth orbit and had the hang of it by Apollo.
Problem 3: Rocket Size (Again). Even with the trick (called Lunar Orbit Rendezvous) used to solve Problem 1, they needed a much bigger rocket than anyone had built before. And so they built it. To get it to the launch pad, they built the crawler transporters, some of the largest land vehicles built up to that time, and to have a protected place to stack the stages, they built the Vehicle Assembly Building, one of the largest buildings by open volume in the world. I think seeing a tower the size of a 36-story skyscraper rise into the sky made a lot of people say, "Wow, they solved that problem?" I was too young at the time, but it was the initial uncrewed Apollo 4 launch of the Saturn V that made my dad think, "Huh, they might actually pull this off!"
There are many many more, but it was really the cumulative effect of solving thousands of hard problems that was the big deal.
(Note: "Problem 3" was added later upon rereading. The visible bigness was just so impressive, and it also is a fair reflection of the huge systems engineering demands that others have mentioned.)